Consider an attacking force approaching a planetary body. Said attacking   
   force, and the planetary defenders, do not use lasers. At what altitude should   
   the attacking force operate for maximum advantage?   
      
   Low orbit is heavily unfavourable for the attackers. Orbital velocity is high,   
   and altitude low, allowing defenders to lob simple low-delta-v projectiles   
   into their path. In contrast, the attackers are forced to attack only targets   
   on the far side of the    
   planet, or equip weapons with copious amounts of delta-v to dispatch near-side   
   targets. Response times are ridiculously low, necessitating fast computers and   
   high-thrust engines for dodging.   
      
   High orbit is heavily unfavourable for the defenders. Kinetic interceptors   
   have to be launched at close to escape velocity to even reach the attackers.   
   Attackers can use low-delta-v rockets to deorbit payloads that impact with   
   tremendous velocity.    
   Response times are protracted, leaving lots of time to either dodge or send up   
   countermeasures.   
      
   Where is the middle ground? A place where defenders and attackers are   
   energetically matched in terms of deploying interceptors, and response times   
   are neither too long nor too short?   
      
   Several assumptions for the following calculations:   
      
   1. Planet is airless and nonrotating   
   2. Attacker occupies circular orbits only.   
   3. Defender fires projectiles straight upward - this makes sense because   
   defenders only care about altitude - not velocity. Anything above 1 km/s   
   generally starts becoming a bad day for either combatant   
   4. For the same reason (and ease of calculation), the attackers fire their   
   projectiles backwards such that they drop with zero lateral velocity, straight   
   downwards. This also helps with targeting, since the landing ellipse is very   
   long when the impact    
   point is on the farside, and shrinks to a small circle when the projectile   
   impacts from above.   
   5. Any contact between projectile and target leads to a kill. Obviously not   
   necessarily true for attackers attacking from low orbit or defenders attacking   
   in high orbit, but that's a problem easily solvable by launching a nuke   
   instead.   
   6. Projectiles are impulsively accelerated (a condition that may be   
   approximated by >100g rocket burns)   
      
   M is mass of planet,   
   R_p is planet radius   
   v_launch is speed at which projectiles are launched   
   R is orbital radius of attacker   
      
   For defender:   
   0.5*(v_launch)^2 = GM((1/R_p)-(1/R)) [conservation of energy - kinetic energy   
   at launch converted completely to potential energy at top of trajectory]   
   (v_launch)^2 = 2GM((1/R_p)-(1/R))   
      
   For attacker:   
   v_launch = sqrt(GM/R)   
   (v_launch)^2 = GM/R   
      
   For energy parity, v_launch is equal for attacker and defender:   
      
   GM/R = 2GM((1/R_p)-(1/R))   
   GM/R = 2GM/R_p - 2GM/R   
   3GM/R = 2GM/R_p   
      
   R = 1.5 R_p [!!!]   
      
   Interestingly, this middle ground seems independent of the planet mass. So   
   long as the planet is round, and the attacker orbit is circular, it will   
   always take the same energy for attackers to launch projectiles to intercept   
   defenders as it takes    
   defenders to intercept attackers at 1.5 times the planetary radius!   
      
   Would-be attackers would be advised to not stray below this altitude, at least   
   in the early going, to avoid granting energy superiority to defenders, and   
   would in practice probably operate at 2 to 3 times the altitude to frustrate   
   defensive tactics.    
   Higher orbits are of course easier, but come with the drawback of longer   
   response time. The middle ground provides the shortest response time   
   achievable without handing over massive advantages to the defenders.   
      
   A few points when adapting this to a planet like Earth:   
      
   The atmosphere hinders defenders, who have to counter aerodynamic drag when   
   launching interceptors to the same altitude, but leaves attackers unaffected,   
   since they merely have to deliver the payload to the target. For Earth, the   
   attacker's projectiles    
   hit the upper atmosphere approaching 6500 m/s^-1 if dropped from middle ground   
   of 3000+km, overkill for most bunkers even if half the velocity is wiped by   
   drag. And there's always the option of adding nukes. Defenders are out of   
   luck, because they need    
   the velocity to even deliver the payload to the attackers. If they do though,   
   the attackers will be whanged with kinetic warheads at 6500 m/s^-1. The net   
   result of all this is that the middle ground orbit shrinks, allowing the   
   attacker to draw closer.   
      
   Rotation causes the middle ground envelope, a sphere, to bulge into an oblate   
   spheroid. This favours the defenders, but only slightly.   
      
   The case for the attackers to drop their projectiles straight downwards is   
   even greater in the case of an atmosphere-laden planet, because an elliptical   
   trajectory cuts through the atmosphere, increasing projectile inaccuracy   
   greatly.   
      
   In practice, these trajectories leave the defenders' projectiles dangerously   
   slow at apoapsis, and the attackers' projectiles dangerously slow at release,   
   situations that are not good for reducing enemy response time. To reduce time   
   taken, significant    
   excess upward velocity may be used by defenders, while the attackers may   
   include a downward component of delta-v to their projectiles to speed it on   
   its descent.   
      
   Thoughts?   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)